Write a structured analysis of {{code}} in a {{tone}} tone for {{target_audience}}. The analysis must follow the O(1) optimization framework: identify operations above O(1), analyze optimization potential, design O(1) solutions, verify correctness, and document tradeoffs. The output must include: - Performance Analysis: for each component, provide name, original complexity, optimized complexity, proof, and implementation code. - Bottleneck Identification: list each bottleneck with location, impact, solution, code, and verification. - Optimization Roadmap: present stages with changes, expected impact, implementation, and verification. The analysis must reflect five iterative passes: identify operations above O(1), analyze optimization potential, design O(1) solutions, verify correctness, document tradeoffs. Constraints: maximum 1500 words. Every complexity claim must have step-by-step proof. All optimization suggestions must include concrete code. If a component cannot be optimized to O(1), state best achievable complexity and explain why. Use plain language that {{target_audience}} with no prior knowledge of the code can understand. Paste any...